The field of the invention is medical imaging and particularly, methods for reconstructing images from acquired image data.
In a computed tomography system, an x-ray source projects a fan-shaped beam which is collimated to lie within an x-y plane of a Cartesian coordinate system, termed the “image plane.” The x-ray beam passes through the object being imaged, such as a medical patient, and impinges upon an array of radiation detectors. The intensity of the transmitted radiation is dependent upon the attenuation of the x-ray beam by the object and each detector produces a separate electrical signal that is a measurement of the beam attenuation. The attenuation measurements from all the detectors are acquired separately to produce what is called the “transmission profile,” “attenuation profile,” or “projection.”
The source and detector array in a conventional CT system are rotated on a gantry within the imaging plane and around the object so that the angle at which the x-ray beam intersects the object constantly changes. The transmission profile from the detector array at a given angle is referred to as a “view,” or “projection,” and a “scan” of the object comprises a set of views made at different angular orientations during one revolution of the x-ray source and detector. In a “cone beam” arrangement, the focal spot of the x-ray source and the detector define a cone-shaped beam of x-rays. When a subject is not fully covered by the cone beam, the views contained therein are said to be “truncated.” The degree of this truncation depends on factors including the size of the detector utilized, the size of the subject, and the view angle. When the subject is a human body, measuring non-truncated cone beam projections requires an impracticably large detector. Thus, in medical applications, the measured cone beam projections are typically always truncated.
In a 2D scan, data is processed to construct an image that corresponds to a two dimensional slice taken through the object. The prevailing method for reconstructing an image from 2D data is referred to in the art as the filtered backprojection technique. This image reconstruction process converts the attenuation measurements acquired during a scan into integers called “CT numbers” or “Hounsfield units,” which are used to control the brightness of a corresponding pixel on a display. The filtered backprojection image reconstruction method is the most common technique used to reconstruct CT images from acquired transmission profiles.
According to standard image reconstruction theories, in order to reconstruct an image without aliasing artifacts, the sampling rate employed to acquire image data must satisfy the so-called Nyquist criterion, which is set forth in the Nyquist-Shannon sampling theorem. Moreover, in standard image reconstruction theories, no specific prior information about the image is needed. On the other hand, when some prior information about the desired image is available and appropriately incorporated into the image reconstruction procedure, an image can be accurately reconstructed even if the Nyquist criterion is violated. For example, if one knows a desired image is circularly symmetric and spatially uniform, only one view of parallel-beam projections (i.e., one projection view) is needed to accurately reconstruct the linear attenuation coefficient of the object. As another example, if one knows that a desired image consists of only a single point, then only two orthogonal projections that intersect at the point are needed to accurately reconstruct an image of the point. Thus, if prior information is known about the desired image, such as if the desired image is a set of sparsely distributed points, it can be reconstructed from a set of data that was acquired in a manner that does not satisfy the Nyquist criterion. Put more generally, knowledge about the sparsity of the desired image can be employed to relax the Nyquist criterion; however, it is a highly nontrivial task to generalize these arguments to formulate a rigorous image reconstruction theory.
Recently, a new mathematical framework for data processing termed “compressed sensing” (CS) has been formulated. Using compressed sensing, only a small set of linear projections of a sparse image are required to reconstruct a quality image. The theory of CS is described by E. Candès, J. Romberg, and T. Tao, in “Robust uncertainty principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information,” IEEE Transactions on Information Theory 2006; 52:489-509, and by D. Donoho in “Compressed Sensing,” IEEE Transactions on Information Theory 2006; 52:1289-1306, and is disclosed, for example, in U.S. patent application Ser. No. 11/199,675.
Although the mathematical framework of CS is elegant, the applicability of CS to a reconstruction method in the field of medical imaging critically relies on the medical images being sparse. Unfortunately, a medical image is often not sparse in the standard pixel representation. Despite this, mathematical transforms can be applied to a single image in order to sparsify the image. Such transforms are thus referred to as “sparsifying transforms.” More specifically, given a sparsifying transform, Ψ, CS image reconstruction can be implemented by minimizing the following objective function:∥ΨI∥1 such that AI=Y  Eqn. (1).
In the above objective function, I is a vector that represents the desired image, Y is a vector that represents the data acquired by the imaging system, A is a system matrix that describes the measurements, and the following:
                                                                      x                                      1                    =                                    ∑                              i                =                1                            N                        ⁢                                                        x                i                                                                  ;                            Eqn        .                                  ⁢                  (          2          )                    
is the so-called L1-norm of an N-dimensional vector, X. Namely, CS image reconstruction determines an image that minimizes the L1-norm of the sparsified image among all images that are consistent with the physical measurements, AI=Y.
The basic ideas in the CS image reconstruction theory can be summarized as follows. Instead of directly reconstructing a desired image in pixel representation, a sparsified version of the desired image is reconstructed. In the sparsified image, substantially fewer image pixels have significant image values; thus, it is possible to reconstruct the sparsified image from an undersampled data set. After the sparsified desired image is reconstructed, an “inverse sparsifying transform” is used to transform the sparsified image back to the desired image. In practice, there is no need to have an explicit form for the “inverse” sparsifying transform. In fact, only the sparsifying transform is needed in image reconstruction.
In practical interventional cardiology, a C-arm system with a small flat-panel detector is used. For example, the panel size is typically only about 20 cm by 20 cm, which provides a field-of-view on the order of only 12 cm. For x-ray projection imaging, this field-of-view is barely sufficient to cover the entire heart. Moreover, each cone beam projection acquired using such a C-arm system is truncated. Even without ECG gating, filtered backprojection reconstruction algorithms can not generally reconstruct a satisfactory image. In fact, it is inherently difficult to reconstruct an image in this situation, provided that no a priori information is available. Therefore, it is even a tremendous challenge to produce a suitable prior image for use in image reconstruction methods, such as prior image constrained compressed sensing (“PICCS”) methods as applied to cardiac imaging.